Place butter and chocolate in a small saucepan over low heat and stir until melted and smooth. Set aside.
In a large bowl, combine the sugar, flour, cocoa, baking powder and salt. Add the vanilla, egg and the butter+chocolate mixture and mix with a spatula or wooden spoon until a smooth dough forms. Turn dough onto a large piece of baking paper and shape into a long cylinder – like Martha does here. Twist the ends and refrigerate until firm, 2-3 hours.
Preheat the oven to 180°C/350°F. Line a large baking sheet with baking paper.
Remove the dough cylinder from the baking paper and slice into 5mm thick slices. Place onto prepared baking sheet 2.5cm (1 in) apart. Bake for 8-10 minutes or until the cookies begin to firm up.
Cool completely on the sheet over a wire rack.
Dip half of each cookie in white chocolate and place onto a rack or piece of baking paper to set – be gently when removing the cookies from the rack or paper because the white chocolate coating might stick to it.